Closed shell system

Total spin den sity reflects th e excess probability of fin din g a versus P electrons in an open-shell system. Tor a system m which the a electron density is equal to the P electron density (for example, a closed-shell system), the spin density is zero. 

Closed-shell Systems and the Roothaan-Hall Equations... 

We shall initially consider a closed-shell system with N electroris in N/2 orbitals. The derivation of the Hartree-Fock equations for such a system was first proposed by Roothaan [Roothaan 1951] and (independently) by Hall [Hall 1951]. The resulting equations are known as the Roothaan equations or the Roothaan-Hall equations. Unlike the integro-differential form of the Hartree-Fock equations. Equation (2.124), Roothaan and Hall recast the equations in matrix form, which can be solved using standard techniques and can be applied to systems of any geometry. We shall identify the major steps in the Roothaan approach. 

The Fock matrix elements for a closed-shell system can be expanded as follows by substituting the expression for the Fock operator ... 

P is the total spinless density matrix (P = P + P ) and P is the spin density matrix (P = p" + P ). For a closed-shell system Mayer s definition of the bond order reduces to ... 

In a closed-shell system, P = P) = P and the Fock matrix elements can be obtained by making this substitution. If a basis set containing s, p orbitals is used, then many of the one-centre integrals nominally included in INDO are equal to zero, as are the core elements Specifically, only the following one-centre, two-electron integrals are non-zero (/x/x /x/x), (pit w) and (fti/lfM/). The elements of the Fock matrix that are affected can then be written a." Uxllow s ... 

The sum of the zeroth-order and first-order energies thus corresponds to the Hartree-Fock energy (compare with Equation (2.110), which gives the equivalent result for a closed-shell system) ... 

Here we give the molecule specification in Cartesian coordinates. The route section specifies a single point energy calculation at the Hartree-Fock level, using the 6-31G(d) basis set. We ve specified a restricted Hartree-Fock calculation (via the R prepended to the HF procedure keyword) because this is a closed shell system. We ve also requested that information about the molecular orbitals be included in the output with Pop=Reg. 

F is called the Fock matrix, and it represents the average effects of the field of all the electrons on each orbital. For a closed shell system, its elements are ... 

Once again, this doesn t help us to find the HF orbitals, because we have to know the LCAO coefficients before we know the Hamiltonian matrix. For the record, and because we will refer to them many times, the elements of the HF matrix for a closed shell system are... 

Multiplying from the left by a specific basis function and integrating yields the Roothaan-Hall equations (for a closed shell system). These are the Fock equations in the atomic orbital basis, and all the M equations may be collected in a matrix notation. 

So far there have not been any restrictions on the MOs used to build the determinantal trial wave function. The Slater determinant has been written in terms of spinorbitals, eq. (3.20), being products of a spatial orbital times a spin function (a or /3). If there are no restrictions on the form of the spatial orbitals, the trial function is an Unrestricted Hartree-Fock (UHF) wave function. The term Different Orbitals for Different Spins (DODS) is also sometimes used. If the interest is in systems with an even number of electrons and a singlet type of wave function (a closed shell system), the restriction that each spatial orbital should have two electrons, one with a and one with /3 spin, is normally made. Such wave functions are known as Restricted Hartree-Fock (RHF). Open-shell systems may also be described by restricted type wave functions, where the spatial part of the doubly occupied orbitals is forced to be the same this is known as Restricted Open-shell Hartree-Fock (ROHF). For open-shell species a UHF treatment leads to well-defined orbital energies, which may be interpreted as ionization potentials. Section 3.4. For an ROHF wave function it is not possible to chose a unitary transformation which makes the matrix of Lagrange multipliers in eq. (3.40) diagonal, and orbital energies from an ROHF wave function are consequently not uniquely defined, and cannot be equated to ionization potentials by a Koopman type argument. 

From the above it should be clear that UHF wave functions which are spin contaminated (more than a few percent deviation of (S ) from the theoretical value of S S + 1)) have disadvantages. For closed-shell systems an RHF procedure is therefore normally preferred. For open-shell systems, however, the UHF method has been heavily used. It is possible to use an ROHF type wave function for open-shell systems, but this leads to computational procedures which are somewhat more complicated than for the UHF case when electron correlation is introduced. 

For closed-shell systems LSDA is equal to LDA, and since this is the most common case, LDA is often used interchangeably with LSDA, although this is not true in the general case (eqs. (6.16) and (6.17)). The method proposed by Slater in 1951 can be considered as an LDA mediod where die correlation energy is neglected and the exchange term is given as... 

Conventional presentaticsis of DFT start with pure states but sooner w later encounter mixed states and d sities (ensemble densities is the usual formulation in the DFT literature) as well. These arise, for example in formation or breaking of chemical bonds and in treatments of so-called static correlation (situations in which several different one-electron configurations are nearly degenerate). Much of the DFT literature treats these problems by extension and generalization from pure state, closed shell system results. A more inclusively systematic treatment is preferable. Therefore, the first task is to obtain the Time-Dependent Variational Principle (TDVP) in a form which includes mixed states. 

At least for the case of a non-degenerate ground state of a closed shell system, it is possible to delineate the standard Kohn-Sham procedure quite sharply. (The caveat is directed toward issues of degeneracy at the Fermi level, fractional occupation, continuous non-integer electron number, and the like. In many but of course not all works, these aspects of the theory seem to be... 

Radicals can be prepared from closed-shell systems by adding or removing one electron or by a dissociative fission. Generally speaking, the electron addition or abstraction can be performed with any system, the ionization potential and electron affinity being thermodynamic measures of the probability with which these processes should proceed. Thus, to accomplish this electron transfer, a sufficiently powerful electron donor or acceptor (low ionization potential and high electron affinity, respectively) is required. If the process does not proceed in the gas phase, a suitable solvent may succeed. 

The F matrix elements in eqs. (15) and (16) are formally the same as for closed-shell systems, the only difference being the definition of the density matrix in eq. (17), where the singly occupied orbital (m) has also to be taken into account. The total electronic energy (not including core-core repulsions) is given by... 

The description of configuration interaction given for rr-electron methods is also valid for all-valence-electron methods. Recently, two papers were published in which the half-electron method was combined with a modified CNDO method (69) and the MINDO/2 method was combined with the Roothaan method (70). Appropriate semiempirical parameters and applications of all-valence-electron methods are most probably the same as those reviewed for closed-shell systems (71). 

These methods can give us useful information on radicals in a manner similar to that for closed-shell systems, provided the exploitation is correct. Of course, in expressions for total energy, bond orders, etc., a singly occupied orbital must be taken into account. One should be aware of areas where the simple methods give qualitatively incorrect pictures. The HMO method, for example, cannot estimate negative spin densities or disproportionation equilibria. On the other hand, esr spectra of thousands of radicals and radical ions have been interpreted successfully with HMO. On the basis of HMO orbital energies and MO symmetry... 

The exploitation, which is, of course, the same as with closed-shell systems (86), is based on the two following plausible assumptions ... 

Eq. (90) can be called an extension of Koopmans theorem to radicals. Recently, Richards (107) discussed the approximate nature of Koopmans theorem in treatments of closed-shell systems, and most probably his arguments apply here also. 

Until now, applications of semiempirical all-valence-electron methods have been rare, although the experimental data for a series of alkyl radicals are available (108,109). In Figure 9, we present the theoretical values of ionization potentials calculated (68) for formyl radical by the CNDO version of Del Bene and Jaffe (110), which is superior to the standard CNDO/2 method in estimation of ionization potentials of closed-shell systems (111). The first ionization potential is seen, in Figure 9, to agree fairly well with the experimental value. Similarly, good results were also obtained (113) with some other radicals (Table VII). 

A detailed diseussion of these and other variants was given in (Ref.35). Attention must be called to the fact that these methods are not variational which causes the energies obtained with them to be lower than those obtained with the FCI method. The counterpart to this deffect is that excited states, open-shell systems, and radicals, can be calculated with as much ease as the ground state and closed-shell systems. Also, the size of the calculation is determined solely by the size of the Hilbert subspace chosen and does not depend in principle on the number of electrons since all happens as if only two electrons were considered. 

Momentum space equations for a closed-shell system... 

Closed-shell systems as defined in the standard Hartree-Foek theory [39-40],... 

In this contribution our purpose is to review the principles and the results of the momentum space approach for quantum chemistry calculations of molecules and polymers. To avoid unnecessary complications, but without loss of generality, we shall consider in details the case of closed-shell systems. 

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